Definitions are the core of mathematical precision; they come to answer "what is X" in mathematics. Into this category fit questions regarding equivalence of definitions; clarifications regarding complicated definitions; as well questions with purposed definitions for mathematical notions with ...

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Question about definition of some classes of bimodules.

Suppose that we have a ring $R$ and a $R$-$R$ bimodule $M$ such that: For every $r\in R$ and $m\in M$ there exists $r'\in R$ such that $m\cdot r=r'\bullet m.$ Examples of this bimodules can be seen ...
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2answers
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A question in Isomorphism

Let G be a cyclic group. Soppose G and G' are isomorphic groups. Show that G' is also cyclic. Can Someone Solve this pleaase? I have an exam 2 hours later!
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45 views

Confused about the definitions of atom

I went though half a dozen books on measure theory, and it occurs to me that the definition of atom is not particularly unified. Version $1$: A set $E$ in a $\sigma$-algebra $\Sigma$ is called ...
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2answers
72 views

On the canonical isomorphism between $V$ and $V^{**}$

I am trying to understand more about the Bidualspace (or double dual space). The whole idea is that $V$ and $V^{**}$ are canonically isomorphic to one another, which means that they are isomorphic ...
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1answer
18 views

Beginning Proof on functions and Sum of functions

I've been having a hard time with this proof because I do not know where to go from where I am. The proof we were assigned to in class is as follows: Let $f : \mathbb Z \rightarrow \mathbb Z$ be a ...
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1answer
42 views

“internal” definition of complete regularity?

There is something strange (I think) about the complete regularity separation axiom. Consider the definitions. T0 means for every two distinct points there is an open set containing exactly one of ...
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1answer
77 views

What are some practical uses of functions? [closed]

Functions are basically formal equations that relate a set of inputs to output. What are some practical uses for functions and inverse functions?
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92 views

Notation for Partial Functions

Suppose we have sets $f$, $A$ and $B$ such that $f\subset A\times B$ and $\forall x\in A\space \forall y,z\in B: [(x,y)\in f \land (x,z)\in f \implies y=z]$ i.e. $f$ is a partial function mapping ...
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1answer
63 views

Different ways to formally define trigonometric functions

When I first learnt trigonometric functions I was in highschool and obviously the explanation they gave me was mostly intuitive. Now that I have taken my first curse of calculus I learnt a formal ...
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1answer
59 views

Nonorientable surfaces: genus or demigenus?

The genus $g$ of a closed, orientable surface is the maximum number of disjoint simple closed curves that can be drawn on the surface without disconnecting it. In terms of the Euler characteristic, ...
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1answer
22 views

what is the definition of linearly independent subset of abelian group?

WHat is the definition of linearly independent subset of abelian group? I know the concept of this, but i don't know how to define this term precisely. Below is what i tried to formulate: Let ...
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1answer
47 views

Linear maps and matrix coefficients

I am currently working through this page in my script: Can somebody explain what this means and how it works in practice? Perhaps if I saw an example I could follow it. Thanks for your help!
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1answer
35 views

What is the definition of “disjoint cycles”?

I'm the one who thinks clear definition(clear with meta-language) is very important for doing mathematics. Below, i list my definitions for cycle and orbit. Let $X$ be a nonempty set. Let ...
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2answers
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What is “the orbit of a permutation”? Is the term “orbit” consistent with that for group action?

reference: What is the orbit of a permutation? To be honest, i don't understand the answer in the link. The orbit of a group action is defined as follows: Let $G$ be a group acting on a set $X$. ...
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3answers
114 views

Integral definition of e

I know that $e$ can be defined via a convergent series: $$ e = \sum_{n=0}^\infty {1\over n!}$$ Or as a limit: $$ e = \lim_{n \to \infty} { \left(1 + {1 \over n}\right)^n }$$ Or as the value which ...
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3answers
385 views

A doubt in the rigorous definition of limits.

I studied the definition of limits today, and I think I mostly understood it, but I have a little doubt. In the definition: $f(x)$ is defined on some open interval containing $a$, except at possibly ...
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1answer
44 views

Lipschitz continuity of $f(x,y)=4x^2+xy-\frac{1}{y-1}$ on an open set $U \subset \mathbb{R} \times (\mathbb{R} \setminus \lbrace 1 \rbrace)$

Problem: Find an open set $U \subset \mathbb{R} \times (\mathbb{R} \setminus \lbrace 1 \rbrace )$ which includes the points $(0, 1/2$) and $(0,3/2)$ such that the function ...
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1answer
22 views

Definition of an integrand

General Question: Say we have integral $$ \int f(z)\ dz $$ Is the integrand in this context (i) $f(z)$ or (ii) $f(z)\ dz$? In any case, is $f(z)\ dz$ a formally defined mathematical object in its ...
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1answer
36 views

Definition of tangent vector

I have a small bit of confusion with the definition my text is providing me with for a tangent vector. Given a manifold $M$, it is first stated that to define a tangent vector, a curve $c:(a,b) ...
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2answers
47 views

Notation - Transpose of Block Matrices [Lay P121 Q2.4.12]

Definition of Transpose is $(A^T)_{ij} = A_{ji}$ $1.$ Why $\begin{bmatrix} M & N \end{bmatrix}^T = \begin{bmatrix} M^T \\ N^T \end{bmatrix}$, and NOT $\begin{bmatrix} M \\ N\end{bmatrix}$? ...
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0answers
26 views

What is the definition of a norm in the context of rings?

On several places in ring-theory I encountered so-called 'norms'. For instance on integral domain $\mathbb{Z}\left[i\right]$ with prescription $a+bi\mapsto a^{2}+b^{2}$ where it also serves as a ...
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0answers
36 views

The term $rank$ in methematics

Reading wikipedia's disambiguation page about the "rank" word I see many concept of rank of many different matematical object. I only know about the rank of a graded poset and the rank of a set that ...
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1answer
46 views

What is the meaning of an algebra?

An algebra $A(*,\hat{} ,\sim)$ is said to be Boolean algebra if it satisfies some conditions...In this statement what is the meaning of starting word an algebra?
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141 views

What is the “Principle of permanence”?

While reading the book "The Number-System of Algebra (2nd edition)." term "Principle of permanence" occurred to me. I remember I had read this in the book "Beginning algebra for college students.". I ...
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2answers
212 views

The phrases “has … in ” vs. “contains … of” in Baby Rudin

Consider the following two statements. (Assume $E \subseteq K$.) $E$ has a limit point in $K$. vs. $E$ contains a limit point of $K$. What do they each mean and how are they different?
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1answer
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What is the formal meaning of “determine” in Baby Rudin 2.40?

In Theorem 2.40, Rudin talks about a $k$-cell $I$ formed by the intervals $[a_1, b_1], \ldots, [a_k, b_k]$. We split each interval at its midpoint $c_j = \frac{a_j + b_j}{2}$ and end up with $2k$ ...
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3answers
144 views

Precise definition of a “game of incomplete information” (Game Theory)

Question: In game theory, what is the precise definition of a "game of incomplete information"? What I've found so far: In the standard first year graduate economics textbook on microeconomics ...
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2answers
69 views

Tensor product is o?

$K$ generated by $(ar,rb)$ for every $a\in A$, $b\in B$, $r\in R$. $F$ generated by $r(a,b)$ for every $a\in A$, $b\in B$, $r\in R$. $(a,rb)=(ar,b)=r(a,b)$,then $K=F$, $F/K=0$ Apparently I ...
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1answer
84 views

Definition of Derivative for $sgn(x)$

When using the definition of the second derivative for $sgn(x)$, I'm a little confused on evaluating something like $sgn(x+h)$. Since $h\rightarrow 0$ does that mean that I should treating $sgn(x+h)$ ...
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6answers
131 views

Orthogonal Projections: Definition + Characterization

Hey there I'm hanging at so many little steps when proving: $P\text{ orthogonal projection}\iff X\text{ orthogonal decomposable}$ My problem is there is simply missing a precise definition of what an ...
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2answers
50 views

What is the definition of 'line' in $\hat{\mathbb{C}}$?

What is the definition of straight line in $\hat{\mathbb{C}}$? Is it defined as $\{x\in\mathbb{C}: \frac{Re(x-a)}{Re(b)} = \frac{Im(x-a)}{Im(b)}\}\cup \{\infty\}$? ($a,b$ are complex numbers and ...
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2answers
66 views

Odd and even functions.

I have a book which says: If a function $f$ satisfies $f(-x)=f(x)$ for all $x$ in its domain, then $f$ is called an even function. However, if $f(-x)=-f(x)$ for every $x$ in the domain of $f$, ...
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3answers
83 views

Orthonormal Bases

I am struggling to get my head around orthonormal bases, this is the defintion in my course notes: If anyone could clarify/explain the concept to me, it would be much appreciated. I am a university ...
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2answers
32 views

Rank of a Matrix under certain conditions

I am a little confused about the rank of a matrix. When does the rank of a matrix equals to zero? Is rank of a matrix equal to zero when it is a zero matrix or the matrix has no elements in it? Thank ...
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0answers
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Computing the differential (not the Jacobi-Matrix) independent of a basis choice

I am eager to learn more about the differential. I was told that it is a good practice to compute the differential independent of a choice of a basis using the following definition Definition: ...
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1answer
65 views

The differential is NOT the Jacobi Matrix?

In the book Analysis II by C.T. Michaels the differential is introduced as the Jacobi-Matrix. In class we had the following definition: Definition: Let $U \subset \mathbb{R}^m$ be open, $f: U \to ...
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1answer
24 views

Derivatives from the left and right

I need to give a definition of the derivative from right and from left. (I know it doesn't make a whole lot of sense, but it is supposed to be similar to the same thing as the limits from the right ...
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2answers
27 views

Homomorphisms, Linear Transformations, and Distributivity

What are the differences between homomorphisms, linear transformations, and distributive operations? To me, they all seem essentially the same, they just are different names for the same phenomenon ...
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1answer
24 views

A question about the span of functions.

I am having trouble understanding the span of functions, my problem is: What is $\operatorname{span}\{a\sin(x),b\}$, and what is its dimension? I understand this in terms of vectors, but not in ...
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1answer
22 views

Can't remember definition of $\lvert G \rvert_{p'}$

For $G$ a finite group, I know that $\lvert G\rvert$ denotes the order of the group. My question is: What is $\lvert G\rvert_{p'}$? Also is this the same as $\lvert G\rvert_p$ (without the prime on ...
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1answer
13 views

Writing order of geospacial coordinates

Is there a defined order in which to write geospacial coordinates? When looking at GPX files generated by my GPS, latitude is named first but when looking at a KML file generated by google earth ...
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1answer
25 views

Approximation Theory: Iterative Methods

Can someone explain to me the general idea of what's going on? I don't get how the functions are being formed, nor do I know why we redefine $h(x)$ to ensure convergence after we defined $g$ ...
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2answers
116 views

If a finite poset has greatest and least elements, is it a lattice?

Let $P$ be a finite partially ordered set with elements $0$ and $1$ such that $0 \le x \le 1$ for any $x \in P$. Does it follow that $P$ is a lattice? If not, what is a counterexample? I believe this ...
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2answers
58 views

What is the definition for holomorphic functions on the Riemann sphere?

I'm trying to study complex analysis in general setting, but i have troubles with defining things in this general setting. I have skimmed definitions in Ahlfors, Conway and wikipedia, but these all ...
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420 views

Definition of set.

A set is defined as a collection of distinct objects. Why have we defined a set to contain only distinct objects? Why is a collection of objects which may have identical objects not called a set? ...
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1answer
38 views

Equivalent definitions for strictly positive elements

We have two usual definitions for strictly positive elements in C*-algebras: Let $A$ be a C*-algebra Definition (a) [MURPHY, C$^*$-algebras and Operator Theory] An element $a\in A_+$ is said to be ...
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4answers
413 views

What is the definition of a positive integer?

I am reading the book "The Number-System of Algebra (2nd edition)". At the starting of page-4 the author writes: A positive integer is a symbol for the number of things in a group of distinct ...
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1answer
67 views

What is this metric called?

Ahlfors -complex analysis p.20 Consider a stereographic projection between the 2-sphere and $\overline{\mathbb{C}}$ (i.e. one-point compactification of $\mathbb{C}$) Let $z,w$ be complex numbers. ...
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1answer
42 views

A question on Holder spaces

A function $f$ is said to belong to the Holder space if Holder condition is satisfied, i.e. $\exists \beta,L\geq0$ such that $$|f(x)-f(x')|\leq L|x-x'|^\beta$$ for all $x,x'$ in the domain of $f$. ...
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1answer
24 views

Why do we define product of morphisms in this way

I'm always forget the definition of product of morphisms in a category, maybe the main reason is because I don't know the motivation beyond the definition: I need help to see this abstract ...